Institute for Systems Research

First, we consider the problem of computing the controllability energyfunction without numerically solving the family of optimal control problems,or the associated Hamilton-Jacobi-Bellman equation, implied in its definition.Stochastically excited systems play a major role in our methodology. Wepresent a stochastic method for computing an estimate of the controllabilityfunction, and show that in certain situations the method provides an exactsolution. The procedure is tested on applications via Monte-Carlo experiments.

Then, we address the problem of numerically determining a Morse transformationfor a function with non-degenerate critical point at 0. We develop analgorithm for computing the desired nonlinear transformation and estimatingthe neighborhood on which the transformed controllability function isquadratic.

In the literature, examples of applied nonlinear balancing have been limited topseudo-balancing of 2-dimensional gradient systems and noting that in the caseof linear systems the energy functions approach reduces to the usual setting ofgramians. We apply our approach to numerically derive, for the first time,balanced representations of nonlinear state-space models. In particular, wepresent applications to a forced damped pendulum system and a forced dampeddouble pendulum system.

In this research, an ULVAC ERA-1000 selective tungsten chemical vapor deposition system located at the University of Maryland was studied where a temperature difference as large as 120 ^oC between the system wafer temperature reading and the thermocoupled instrumented wafer measurement was found during the manual processing mode.

The goal of this research was to develop a simplified, but accurate, three-dimensional transport model that is capable of describing the observed reactor behavior.

A hybrid approach combining experimental and simulation studies was used for model development. Several sets of experiments were conducted to investigate the effects of process parameters on wafer temperature.

A three-dimensional gas flow and temperature model was developed and used to compute the energy transferred across the gas/wafer interface. System dependent heat transfer parameters were formulated as a nonlinear parameter estimation problem and identified using experimental measurements.

Good agreement was found between the steady-state wafer temperature predictions and experimental data at various gas compositions, and the wafer temperature dynamics were successfully predicted using a temperature model considering the energy exchanges between the thermocouple, wafer, and showerhead.

We offer a method for computing a workingapproximation to the controllability energy function, one of the mainobjects involved in the method. Moreover, we show that for a class ofsecond-order mechanical systems with dissipation, under certain conditionsrelated to the dissipation, an exact formula for the controllabilityfunction can be derived. We then present an algorithm for a numericalimplementation of the Morse-Palais lemma, which produces a local coordinatetransformation under which a real-valued function with a non-degeneratecritical point is quadratic on a neighborhood of the critical point.

Application of the algorithm to the controllabilty function plays a key rolein computing the balanced representation. We then apply our methods andalgorithms to derive balanced realizations for nonlinear state-space modelsof two example mechanical systems: a simple pendulum and a double pendulum.

The second part (Chapter 5) deals with modeling of rapid thermal chemicalvapor deposition (RTCVD) for growth of silicon thin films, viafirst-principles and empirical analysis. We develop detailedprocess-equipment models and study the factors that influence depositionuniformity, such as temperature, pressure, and precursor gas flow rates,through analysis of experimental and simulation results. We demonstratethat temperature uniformity does not guarantee deposition thicknessuniformity in a particular commercial RTCVD reactor of interest.

In thethird part (Chapter 6) we continue the modeling effort, specializing to acontrol system for RTCVD heat transfer. We then develop and apply ad-hocversions of prominent model reduction approaches to derive reduced modelsand perform a comparative study.

(This work to appear in Proc. 1999 American Control Conference.)

Previous and concurrent research in the modeling of magnetostrictive actuators and the related area of electrostrictive actuators have produced models that are of low order and reproduce their quasi-static response reasonably well. But the main interest in using these and other smart actuators is at a high frequency -- for producing large displacements with mechanical rectification, producing sonar signals etc. The well known limitation of smart actuators that are based on electro-magneto-thermo-elastic behaviors of smart materials is the complex, input-rate dependent, hysteretic behavior of the latter.

The model proposed in this dissertation is a bulk model and describes the behaviour of a magnetostrictive actuator by a system with 4 states. We develop this model using phenomenological arguments following the work done by Jiles and Atherton for describing bulk ferromagnetic hysteresis. The model accounts for magnetic hysteresis; eddy current effects; magneto-elastic effects; inertial effects; and mechanical damping. We show rigorously that the system with the intial state at the origin has a periodic orbit as its $Omega$ limit set. For the bulk ferromagnetic hysteresis model - a simplification of the magnetostrictive model, we show that all trajectories starting within a certain set approach this limit set.

It is envisioned that the model will help application engineers to do simulation studies of structures with magnetostrictive actuators. Towards this end, an algorithm is proposed to identify the various parameters in the model.

In control applications, one may require the actuator to follow a certain trajectory. The complex rate dependent behaviour of the actuator makes the design of a suitable control law a challenging one. As our system of equations do not model transient effects, they do not model the minor-loop closure property common to ferromagnetic materials. Therefore, the design of control laws making explicit use of the model (without modifications) is not possible.

A major reason to use model free approaches to control design is that magnetostrictive actuators seem to have slight variations in their behavior with time. Therefore, we tried to use a direct adaptive control methodology that uses features of our model. The system is now looked at as a relative degree two linear system with set-valued input nonlinearity. Extensions of Eugene Ryan's work on universal tracking for a relative degree one linear system and Morse's work on stablization for relative degree two linear systems were sought.

Experimental verification of our method confirmed our intuition about the model structure. Though the tracking results were not very satisfactory due to the presence of sensor noise, the experimental results, nevertheless validate our modeling effort.

Bothpitchfork bifurcation and transcritical bifurcation are addressed.The results obtained forpitchfork bifurcations apply to general nonlinear models smoothin the state and the control. For transcritical bifurcations,the results require the system to be affine in the control.

Using this approach, both current mode control and discontinuous conduction modecan be handled systematically in a unified framework,making the modeling for these cases simpler than with the use of averaging.The results of this paper are similar to the results of Tymerski,but they are presented in a simpler manner tailored to facilitate immediate application to specific circuits.

It is shown howsampling the output at certain instants improves the obtained phase response.Frequency responses obtained from the sampled-data model aremore accurate than those obtained from various averaged models. In addition, a new ("lifted")continuous-time switching frequency-dependent model of the power stage isderived from the sampled-data model. Detailed examples illustrate themodeling tools presented here, and also provide a means of comparingresults obtained from the sampled-data approach with those obtainedfrom averaging.

General models are presented in a unified and simple manner, while removingsimplifying approximations present in previous work. These models applyboth for current mode control and voltage mode control.The general models are nonlinear. They are used toobtain {it analytical} linearized models, which are in turn employedto obtain local stability results.

Detailed examplesillustrate the modeling and analysis in the paper,and point to situations in which the sampled-data approachgives results superior to alternate methods.For instance, it is shownthat the sampled-data approach will reliablypredict the (local) stability of aconverter for which averaging or simulation predicts instability.